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Sparse subspace clustering via nonconvex approximation

PATTERN ANALYSIS AND APPLICATIONS 2019年第1期22卷 165-176页

作者： Dong, Wenhua Wu, Xiao-Jun Kittler, Josef Yin, He-Feng Jiangnan Univ Sch Internet Things Wuxi 214122 Peoples R China Univ Surrey Ctr Vis Speech & Signal Proc Guildford GU2 7XH Surrey England

Among existing clustering methods, sparse subspace clustering (SSC) obtains superior clustering performance in grouping data points from a union of subspaces by solving a relaxed 0-minimization problem by 1-norm. The use of 1-norm instead of the 0 one can make the object function convex, while it also causes large errors on large coefficients in some cases. In this work, we propose using the nonconvex approximation to replace 0-norm for SSC, termed as SSC via nonconvex approximation (SSCNA), and develop a novel clustering algorithm with the enhanced sparsity based on the Alternating Direction Method of Multipliers. We further prove that the proposed nonconvex approximation is closer to 0-norm than the 1 one and is bounded by 0-norm. Numerical studies show that the proposed nonconvex approximation helps to improve clustering performance. We also theoretically verify the convergence of the proposed algorithm with a three-variable objective function. Extensive experiments on four benchmark datasets demonstrate the effectiveness of the proposed method.

关键词： Sparse subspace clustering 0-minimization nonconvex approximation 1-norm

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Hyperspectral Image Denoising via Correntropy-Based nonconvex Low-Rank approximation

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IEEE JOURNAL OF SELECTED TOPICS IN APPLIED EARTH OBSERVATIONS AND REMOTE SENSING 2024年 17卷 6841-6859页

作者： Lin, Peizeng Sun, Lei Wu, Yaochen Ruan, Weiyong Sun Yat sen Univ Sch Syst Sci & Engn Guangzhou 510006 Peoples R China

Hyperspectral images (HSIs) are prone to be corrupted by various types of noise during the process of imaging and transmission, which seriously affect the subsequent HSI processing tasks. In this article, we proposed a novel low-rank-based model for HSIs denoising. On one hand, motivated by the superiority of nonconvex approximation to matrix rank, we construct a new nonconvex function which is tighter than some of existing rank approximated functions. On the other hand, we describe Gaussian noise and sparse noise simultaneously by introducing correntropy. In comparison with traditional model, we constrain noise in one regularization instead of separately constraining Gaussian noise and sparse noise, which indicates that the number of regularization parameters have been reduced. To optimize the proposed model, some convex analysis tools are utilized in this article. In addition, we provide theoretical analysis on the convergence of the developed algorithm. Through experiments conducted on both simulated and real HSIs, we verify the superiority of our model in enhancing the performance of mixture noise removal in HSIs.

关键词： Noise reduction Gaussian noise Sparse matrices Training Deep learning Correlation Tensors Correntropy hyperspectral image (HSI) denoising low-rank presentation nonconvex approximation

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nonconvex Optimization Problems for Maximum Hands-Off Control

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL 2025年第3期70卷 1905-1912页

作者： Ikeda, Takuya Univ Kitakyushu Fac Environm Engn Fukuoka 8080135 Japan

Maximum hands-off control is the optimal solution to the L(0 )optimal control problem. While convex approximation is typically used to relax this problem, it does not necessarily result in maximum hands-off control. Therefore, this study introduces a nonconvex approximation method and a class of nonconvex optimal control problems that are always equivalent to the maximum hands-off control problem. A computation method based on difference of convex functions optimization is then derived and numerically validated.

关键词： Optimal control Optimization Vectors Cost function approximation methods Convex functions Sparse approximation State estimation Standards Optimization methods Difference of convex functions nonconvex approximation optimal control sparse control

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Two-stage image segmentation based on nonconvex l₂ - l_p approximation and thresholding

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APPLIED MATHEMATICS AND COMPUTATION 2021年 403卷 126168-126168页

作者： Wu, Tingting Shao, Jinbo Gu, Xiaoyu Ng, Michael K. Zeng, Tieyong Nanjing Univ Posts & Telecommun Sch Sci Nanjing 210023 Peoples R China Univ Hong Kong Dept Math Pokfulam Hong Kong Peoples R China Chinese Univ Hong Kong Dept Math Satin Hong Kong Peoples R China

Image segmentation is of great importance in image processing. In this paper, we propose a two-stage image segmentation strategy based on the nonconvex l(2) - l(p) approximation of the Mumford-Shah (MS) model, where we use the nonconvex l(p) (0 < p < 1) regularizer to approximate the Hausdorff measure and to extract more boundary information. In the first stage, we solve the nonconvex variant of the MS model efficiently via the split-Bregman algorithm. Moreover, we use a closed-form p-shrinkage operator to deal with the l(p) quasi-norm subproblem, which is easy to implement. The second stage is segmenting the u obtained in the first stage into different phases with thresholds determined by the K-means clustering method. We compare our method with several state-of-the-art methods both qualitatively and quantitatively to demonstrate the effectiveness and advantages of our strategy. (C) 2021 Elsevier Inc. All rights reserved.

关键词： Image segmentation Two-stage strategy Split-Bregman nonconvex approximation

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Low-Rank Quaternion approximation for Color Image Processing

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IEEE TRANSACTIONS ON IMAGE PROCESSING 2020年 29卷 1426-1439页

作者： Chen, Yongyong Xiao, Xiaolin Zhou, Yicong Univ Macau Dept Comp & Informat Sci Macau 999078 Peoples R China South China Univ Technol Sch Comp Sci & Engn Guangzhou 510006 Guangdong Peoples R China

Low-rank matrix approximation (LRMA)-based methods have made a great success for grayscale image processing. When handling color images, LRMA either restores each color channel independently using the monochromatic model or processes the concatenation of three color channels using the concatenation model. However, these two schemes may not make full use of the high correlation among RGB channels. To address this issue, we propose a novel low-rank quaternion approximation (LRQA) model. It contains two major components: first, instead of modeling a color image pixel as a scalar in conventional sparse representation and LRMA-based methods, the color image is encoded as a pure quaternion matrix, such that the cross-channel correlation of color channels can be well exploited;second, LRQA imposes the low-rank constraint on the constructed quaternion matrix. To better estimate the singular values of the underlying low-rank quaternion matrix from its noisy observation, a general model for LRQA is proposed based on several nonconvex functions. Extensive evaluations for color image denoising and inpainting tasks verify that LRQA achieves better performance over several state-of-the-art sparse representation and LRMA-based methods in terms of both quantitative metrics and visual quality.

关键词： Low-rank matrix approximation image denoising image inpainting quaternion singular value decomposition nonconvex approximation

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Tensor Robust Principal Component Analysis via Tensor Fibered Rank and lp Minimization

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SIAM JOURNAL ON IMAGING SCIENCES 2023年第1期16卷 423-460页

作者： Gao, Kaixin Huang, Zheng-Hai Tianjin Univ Sch Math Tianjin 300350 Peoples R China

Tensor robust principal component analysis (TRPCA) is an important method to handle high -dimensional data and has been widely used in many areas. In this paper, we mainly focus on the TRPCA problem based on tensor fibered rank for sparse noise removal, which aims to recover the low-fib ered-rank tensor from grossly corrupted observations. Usually, the l1-norm is used as a convex approximation of tensor rank, but it is essentially biased and fails to achieve the best estimation performance. Therefore, we first propose a novel nonconvex model named TRPCAp, in which the lp norm (0 < p < 1) is adopted to approximate tensor fibered rank and measure sparsity. Then, an error bound of the estimator of TRPCAp is established and this error bound can be better than those of similar models based on Tucker rank or tubal rank. Further, we use the alternating direction method of multipliers to solve TRPCAp and provide convergence guarantee. Finally, extensive experiments on color images, videos, and hyperspectral images demonstrate the effectiveness of the proposed method.

关键词： tensor robust principal component analysis nonconvex approximation ADMM tensor fibered rank

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Efficient approaches for l₂-l₀ regularization and applications to feature selection in SVM

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APPLIED INTELLIGENCE 2016年第2期45卷 549-565页

作者： Hoai An Le Thi Tao Pham Dinh Thiao, Mamadou Ton Duc Thang Univ Dept Management Sci & Technol Dev Ho Chi Minh City Vietnam Ton Duc Thang Univ Fac Math Stat Ho Chi Minh City Vietnam Univ Lorraine Lab Theoret & Appl Comp Sci F-57045 Metz France Univ Normandie INSA Rouen Math Lab Ave Univ F-76801 St Etienne France

For solving a class of a"" (2)- a"" (0)- regularized problems we convexify the nonconvex a"" (2)- a"" (0) term with the help of its biconjugate function. The resulting convex pr... 详细信息

For solving a class of a"" (2)- a"" (0)- regularized problems we convexify the nonconvex a"" (2)- a"" (0) term with the help of its biconjugate function. The resulting convex program is explicitly given which possesses a very simple structure and can be handled by convex optimization tools and standard softwares. Furthermore, to exploit simultaneously the advantage of convex and nonconvex approximation approaches, we propose a two phases algorithm in which the convex relaxation is used for the first phase and in the second phase an efficient DCA (Difference of Convex functions Algorithm) based algorithm is performed from the solution given by Phase 1. Applications in the context of feature selection in support vector machine learning are presented with experiments on several synthetic and real-world datasets. Comparative numerical results with standard algorithms show the efficiency the potential of the proposed approaches.

关键词： Sparsity Zero norm Convex relaxation Biconjugate function nonconvex approximation DC programming DCA Feature selection in SVM

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A nonconvex Implementation of Sparse Subspace Clustering: Algorithm and Convergence Analysis

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IEEE ACCESS 2020年 8卷 54741-54750页

作者： Deng, Xiaoge Sun, Tao Du, Peibing Li, Dongsheng Natl Univ Def Technol Coll Comp Natl Lab Parallel & Distributed Proc PDL Changsha 410073 Peoples R China Northwest Inst Nucl Technol Xian 710024 Peoples R China

Subspace clustering has been widely applied to detect meaningful clusters in high-dimensional data spaces. And the sparse subspace clustering (SSC) obtains superior clustering performance by solving a relaxed l(0)-minimization problem with l(1)-norm. Although the use of l(1)-norm instead of the l(0) one can make the object function convex, it causes large errors on large coefficients in some cases. In this paper, we study the sparse subspace clustering algorithm based on a nonconvex modeling formulation. Specifically, we introduce a nonconvex pseudo-norm that makes a better approximation to the l(0)-minimization than the traditional l(1)-minimization framework and consequently finds a better affinity matrix. However, this formulation makes the optimization task challenging due to that the traditional alternating direction method of multipliers (ADMM) encounters troubles in solving the nonconvex subproblems. In view of this, the reweighted techniques are employed in making these subproblems convex and easily solvable. We provide several guarantees to derive the convergence results, which proves that the nonconvex algorithm is globally convergent to a critical point. Experiments on two real-world problems of motion segmentation and face clustering show that our method outperforms state-of-the-art techniques.

关键词： Sparse subspace clustering nonconvex approximation ADMM reweighted algorithms

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DCA based approaches for bi-level variable selection and application for estimate multiple sparse covariance matrices

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NEUROCOMPUTING 2021年 466卷 162-177页

作者： Le Thi, Hoai An Phan, Duy Nhat Dinh, Tao Pham Univ Lorraine Dept IA LGIPM F-57000 Metz France Inst Univ France IUF Paris France Univ Normandie Lab Math INSA Rouen Caen France

Variable selection plays an important role in analyzing high dimensional data and is a fundamental problem in machine learning. When the data possesses certain group structures in which individual variables are also meaningful scientifically, we are naturally interested in selecting important groups as well as important variables within the selected groups. This is referred to as the bi-level variable selection which is much more complex than the selection of individual variables. In recent years, research on the topic of variable selection is very active, but the majority of the work is focused on the individual variable selection. There is therefore a need to further develop more effective approaches for bi-level variable selection. Since DC (Difference of Convex functions) programming and DCA (DC Algorithm), powerful tools in non-convex programming framework, have been successfully investigated for individual variable selection, we believe that they could be efficiently exploited for the more difficult bi-level variable selection task. In that direction, we investigate in this work DC approximations of the mixed zero norm (l(0,0)) and the combined norm (l(0) + l(q,0)). The resulting approximate problems are then formulated as DC programs for which DCA based algorithms are introduced. As an application, these DCA schemes are developed for estimating multiple sparse covariance matrices sharing some common structures such as the locations or weights of non-zero elements. The experimental results on both simulated and real datasets indicate the efficiency of our algorithms. (C) 2021 Elsevier B.V. All rights reserved.

关键词： Bi-level variable selection Mixed norm nonconvex approximation Multiple covariance matrices DC programming DCA

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Low-rank matrix recovery problem minimizing a new ratio of two norms approximating the rank function then using an ADMM-type solver with applications

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2024年 438卷

作者： Gao, Kaixin Huang, Zheng-Hai Guo, Lulu Tianjin Univ Sch Math Tianjin 300350 Peoples R China

Since the nuclear norm may lead to suboptimal solutions to rank minimization problems, many nonconvex surrogates have been proposed to provide better rank function approximations. In this paper, we use the ratio of the nuclear norm and the Frobenius norm, denoted as N/F, as a new nonconvex surrogate of the rank function. The N/F can provide a close approximation for the matrix rank. In particular, the N/F is the same as the rank for rank 1 matrices. Furthermore, compared to other popular nonconvex approximations, the N/F model has no extra parameters to tune. Theoretically, we establish the exact recovery condition and the stable recovery condition for the N/F minimization problem with or without noise, respectively. Computationally, we use the alternating direction method of multipliers with a specific variable-splitting scheme to solve the N/F minimization problem. What is more, we incorporate a box constraint in the N/F model, and propose the ADMM-box to solve it. The convergence analysis of ADMM and ADMM-box are also given. Extensive experiments on both synthetic data and color images verify the effectiveness of our proposed methods.(c) 2023 Elsevier B.V. All rights reserved.

关键词： Low rank matrix recovery nonconvex approximation Nuclear norm Frobenius norm

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